Climate science does not employ the logic of pure mathematics—Comment by a warmist (see below)

“Future generations will wonder in bemused amazement that the early 21st century’s developed world went into hysterical panic over a globally averaged temperature increase of a few tenths of a degree and, on the basis of gross exaggerations of highly exaggerated computer predictions combined into implausible chains of inference, proceeded to contemplate a rollback of the industrial age.” —Richard Lindzen

Evidence is mounting that the Sun may to be headed into a lengthy spell of low activity, leading to into a mini Ice Age. As global warming theory cools, the scientists who once supported this politicized science are redacting their internet traces. The following conversation was rescued just before the warmist (name suppressed) removed his posts from a well known blog.

**Warmist**: However, it has been extremely difficult, to find competing hypotheses that are of equal or surpassing strength to the anthropogenic hypothesis.

**Philip Pennance**: Which anthropogenic hypothesis?

**Warmist**: That our activities, principally those associated with the emission of very large annual delivery of fossil carbon to the atmosphere leading to significant and well-documented increases in atmospheric greenhouse gases, are the primary reason that surface air temperatures have increased as much as they have since the pre-industrial era. It is this hypothesis that the IPCC-AR4 agreed was probable at a minimum confidence of 90%. Some have reported that the Committee agreed on an even higher confidence level (between 95% and 99%), but that political pressure from big, future emitters, like China, forced them to adopt a lower number.**Philip Pennance**: I don’t see how a meaningful probability can be ascribed to such a mathematically vague hypothesis. For example, what is meant by the term “significant temperature rise” in this setting? The very notion seems to be a function of the measurement period. Two scientists using different time intervals will in general obtain different averages for the temperature at a point.

**Warmist**: It’s not really a vague hypothesis, Philip, and generating the probability is difficult but not impossible. This is roughly how they do it. First, list all potential causes of climate change. Then, measure their potential magnitudes and define confidence intervals for those magnitudes using statistical and physical models calibrated against historical data and future projections of certain key factors (such as whether we will continue to raise our emissions rate, or start curbing our usage). Finally, use a well-established form of statistical meta-analysis to determine the overall probability that anthropogenic sources are the primary cause.

How time intervals are defined are very important, and you are right to point them out. For instance, most of the carbon emissions that the human species has released into the atmosphere throughout its history occurred during the last 50 years, and the expected time lag for an effect on climate by those emissions is on the order of decades. As such, we expect the strongest anthropogenic effect to be evident only recently. This sort of information is included in the confidence intervals of the expected effects of each of the potential causes that we would have hypothetically listed above, and without that temporal information, those confidence intervals don’t mean much of anything.

All of this is just one of the many things you would learn if you were take an upper-level undergraduate or graduate course in climate science. Thousands of articles and reviews have been written on the subject, and Chapter 9 of IPCC-AR4 does a pretty good job summarizing them for all of us. Head there if you want a better answer than I could possibly provide. The bottom line, though, is that while you raise an important question, it was the climate scientists who first asked it and are still asking it to this day.

Philip—a “beginner’s” source for all things climate related can be found at Real Climate

**Philip Pennance**: This procedure is not rigorous in any mathematical sense. Many variables (e.g. temperature of a non homogeneous fluid body) are ordinal —which means that they cannot be averaged in a physically meaning way. In fact, the same set of temperature readings can be consistent both with global warming and global cooling depending on the temperature scale employed. Even if a scale is fixed, an increase in average temperature can easily shown to be consistent with cooling (net heat loss) in a non homogeneous body such as the earth. In any case, whether or not there has been an increase in the average of some quantity over time depends on the starting time from which that average is computed. This is an arbitrary decision, which makes the alleged phenomenon observer dependent.

**Warmist** It’s not a purely mathematical exercise—it involves analysis of the physical phenomena, including global inhomogeneities in heating, mass and energy transport, outgoing long-wave radiation, the sites of high rates of photosynthesis (elsewhere called “forests”, and our knowledge of past events and projections of future events, but it also critically includes information about time scale, as I mentioned above. As such, I don’t understand what your concern is. If you could be more specific, maybe citing an example, it would help.

**Philip Pennance**: The statement that in a non homogeneous body such as the earth an increase in average temperature can be consistent with cooling (net heat loss) is not a purely mathematical statement. Neither is the statement that temperature is an ordinal random variable. Both statements derive (via mathematics) from the laws of classical thermodynamics. However, from the second statement, it follows as a corollary that the notion of a mean global surface temperature is physically meaningless. Thus, alleged rises in this parameter should be treated with caution.

You claim that It is this hypothesis that the IPCC-AR4 agreed was probable at a minimum confidence of 90%. Some have reported that the Committee agreed on an even higher confidence level (between 95% and 99%), but that political pressure from big, future emitters, like China, forced them to adopt a lower number.

This is very revealing! So the significance level may have been changed by a factor of 10 (from 0.01 to 0.1). The mere fact that political pressure could alter the result of a computation of such major scientific import, and by so much, suggests that plenty of hand waving, subjective assumptions and “slack” parameters must have entered into the computations. After all, solid scientific and mathematical claims are not so easily subject to such political manipulation which would easily be detected. Thus, in reality, we can have no confidence even in the confidence interval.

**Warmist**: Philip, it seems that you’re preaching scientific materialism to a scientist. So, again, I am left confused.

Are you arguing with me that statistics is a branch of mathematics? No argument here. Of course it is. My argument was that it is not purely statistics that has allowed climate scientists to put error bars on their estimates. Some of these error bars are ranges of potential values based on the range of potential carbon emissions and modeling scenarios, physically founded factors that are included for your edification, not mystification.

Are you arguing that mean values are abstractions? No argument here. Of course temperature varies geographically, and of course surface temperatures are expected to change differently in different locations (one doesn’t need a complex climate model to predict that warming will be maximal at high latitudes and high altitudes—in fact one needs only to know that the equator and low altitudes receive a net influx of long- and short-wave radiation while the poles provide a net efflux of long- and short-wave radiation). Average values are presented as diagnostic summaries, not universally applicable values. Just because the average height of American males is 5’8” or something, doesn’t mean that all American males are 5’8”, and I don’t think you’ll find a climate scientist who would make such an argument about climate.

Are you arguing that heat is not the same thing as temperature? No argument here. One can heat a parcel of air while it is expanding and still observe a drop in temperature. Basic thermodynamics, as you say, but hardly a consideration left by the wayside in climate science. (What I want to know, and asked you about before, is if you know of a concrete, non-abstract example in which this is found to be generally important in climate science.)

Are you arguing that policy and political considerations should not have been allowed to affect the scientific statement of the IPCC AR4 Working Group 1? Again, no argument here. Of course, China and countries like it should never have been allowed to lower the confidence interval that the scientific community was convinced was justified (I should note that friends of mine who still work in climate science—I no longer do—are aware of several countries that pushed the panel to soften their findings). But that’s the nature of the beast: the IPCC is bound not only to make scientific assessments but also to present policy avenues with which everyone can feel satisfied. Hardly an argument in favor denialism. The scientific assessment was watered down by political expediency, not strengthened.

Until you can provide concrete examples of “slack” variables or the other things you mention, or evidence that they went unreported, I must remain utterly confused and unable to comprehend the point you are trying to make.

**Philip Pennance**:

I did not argue that heat is not the same as temperature — this is obvious. Thus your question is moot. I have a great many concerns which hopefully someone can clarify for me. Let me start with the first issue which I raised concerning confidence intervals. The IPPC’s “anthropogenic hypothesis” is a statement concerning an increase in temperature. I presume that the term “temperature” refers to mean global surface temperature.

Consider the following “thought experiment”:

Suppose that a sequence of independent, identically distributed measurements X_1, X_2, …, X_n is made of the length μ of a rigid rod, then, by the central limit theorem, the sampling distribution of the mean of X_1, X_2, …, X_n converges to a normal distribution with mean equal to the common mean of the sample distributions. In the absence of bias, the mean of the sample distribution will be the exact length μ. Moreover, the variance of the data can be compared to that expected from knowledge of experimental errors inherent in the details of the measurement process. Unexpected discrepancies can then be evaluated and possible model weaknesses analyzed or unexpected sources of error sought.

Application of this standard measurement procedure to, for example, the temperature measurement of a homogeneous fluid body in a constant state would quite likely raise few concerns. On the other hand, the surface of the earth is not a homogeneous fluid body in a constant state. If a number of different observers each using different sets of surface points make estimates of the global mean surface temperature, then the sequence of values so obtained does not represent a list of independent, identically distributed random variables and the hypotheses of the central limit theorem do not hold. Rather, this situation is like unto a set of observers, ignorant of the precepts of special relativity, traveling at different, but high, velocities relative to each other, and attempting to ascribe a length to a rigid rod which does not lie in their respective rest frames. Since the model hypothesis of a Galilean spacetime is false, the very concept of length is observer dependent and thus physically meaningless. Differences in length obtained by different observers cannot be attributed merely to technical difficulties connected with the measuring process but rather to the incorrect model assumption of a Galilean spacetime. Due to the relativity of simultaneity, different observers are, in reality, measuring the separations between different pairs of spacetime events. Discrepancies obtained by such observers cannot in this case be ascribed to experimental error which can in principle be overcome by sufficient ingenuity; rather, their explanation requires a fundamental change of philosophy and the recognition that each is measuring a different parameter. In this case, knowledge of the experimental errors involved and of the discrepancies between observers would hopefully lead to a change in the model hypothesis and the discovery of special relativity. This, in turn, would enable the observers to transform their measurements to the rest frame of the rod, thereby yielding consistent estimates of the true rest length.

Unfortunately, the measurement of global mean surface temperature is more complex than the measurement of a length and the discrepancy due to different observers measuring different distributions cannot be so easily transformed away. Various authors attribute such discrepancy to factors such as an uneven distribution of weather stations. This attribution implicitly assumes that such errors are amenable to correction by more uniform sampling or, perhaps, by better “statistical data processing”. However, the onus is on the researcher to prove that this is the case. One can construct (non physical) temperature distributions whose mean cannot be estimated without bias by any finite sample. Does there exist a proof that the earth’s temperature distribution does not share this pathology?

**Warmist**:: Philip, I apologize if I misunderstood previously about what you were asking.

First, I should point out that no one is observing anything in the climate system at relativistic time, space, or energy scales.

Second, to answer your final question, no, I don’t believe there exists any such proof. But if you think that climate scientists avoid reporting the kind of bias you refer to—which most definitely exists and is worth pointing out—then you’re quite mistaken. Since the early 80s, using atmospheric wind data as a function of 3-D space and time (called “tracer” models), many researchers have been using statistical methods based on those winds data to construct maps of where their sampling is least represented, and how not having measurements in certain areas is biasing their estimates of what’s happening generally at a global scale. I was second author on such a paper in 1996, and while I can no longer speak to what exactly we did or how we did it (I have long since moved on to another discipline), I can vouch for the fact that we did in fact perform such an analysis, and that we cited dozens that had performed similar analyses before us. (Our primary result was that sampling in tropics was too infrequent to be able to use measurements of the way that atmospheric CO2 concentrations vary with the seasons to estimate seasonal changes in the photosynthesis of big forests in the tropics.)

If climate scientists weren’t reporting the bias you’re talking about, then they would be a very sorry lot indeed. But the fact is that they can measure the sensitivity of their estimates to this bias, and they have reported it in full on many occasions.

**Philip Pennance**: You say that “no one is observing anything in the climate system at relativistic time, space, or energy scales.” However, as an ANALOGY the above relativistic thought experiment does highlight one (of many) difficulties present in the measurement of average temperature. I apologize if my previous post is not clear in this respect and will attempt to refine the idea.

Let us imagine once more that two observers, with distinct but very large relative velocity, make repeated measurements on the length of the same rigid rod so as to obtain confidence intervals. Assume also that both observers take the utmost possible care to avoid bias in their measurement techniques. After a certain number of measurements their respective confidence intervals for the length will inevitably be disjoint even though both have avoided any procedural bias. This is a simple consequence of the central limit theorem and the fact that, due to relativistic length dilation, the two observers are in effect sampling from distributions with distinct means. Suppose now that one of the observers attempts to reduce this discrepancy by taking additional measurements. No matter how careful he is, or how many measurements are made, the most that can be achieved, will be a reduction in the size of his confidence interval. The problem of disjoint intervals will not disappear. Compare this to a situation in which two climatologists independently measure the average temperature in some region, using different set of sample points, and find a similar unreconcilable discrepancy between their respective results. Since all climatologists are honest, both will dutifully increase the number of sample points in order to obtain, say, a more uniform or, perhaps, less biased sample. Notwithstanding their undoubted honesty, their best efforts and good intentions are doomed to failure. As in the relativistic situation, the distributions which each climatologist samples are distinct, and so the discrepancy cannot be so eliminated by further measurement. In reality, the temperature distribution over the surface of the earth, at any moment, is quite irregular and it seems improbable that two distinct sets of points on the surface taken at different times (or even at the same time) yield samples from the same temperature distribution. In our relativistic analogy, the discrepancy is easily corrected by a transformation to the rest frame of the rod. In the case of temperature, it would seem that one is compelled to make ad hoc assumptions concerning the regularity of the global temperature distribution. Unfortunately, distinct ad hoc assumptions translate into distinct confidence intervals for the average temperature leading to disputes —which seems to have happened

Even to obtain a confidence interval for the (time) average temperature at a single point, assumptions have to be made concerning the temporal regularity of the temperature at the said point. In fact, the very existence an average temperature underlying the observed temporal fluctuations has to be assumed —:after all, not every random variable has a mean. Thus, for the sake of simplicity, let us eliminate this difficulty from the discussion by considering the best possible scenario of a sphere whose surface temperature distribution is constant in time. Unfortunately, even in this case there exist temperature distributions which are intractable to any conceivable sampling procedure. Consider, for example, the case of a distribution which is equal to a constant c over the surface of a sphere except for a single point P at which the temperature is a delta function of weight normalized so that the average temperature over the sphere is exactly equal to β + c where β is non zero. Then any uniform random sample of points on the sphere would, with probability 1, fail to contain the point P and so yield a sample mean temperature equal to c. Thus, the sample mean would differ from the true mean by the “bias” β which could be any pre-prescribed large number. Infinitely many variants of this construction are possible. Moreover, weaker (physical) versions could be constructed without recourse to the theory of distributions. There possibly exists no proof that the earth’s temperature distribution does not share this pathology. On the surface of the earth an infinity of sharp temperature jumps occur in such places as the boundaries of icebergs floating in warmer water, at shorelines which are ever shifting and changing and, indeed, at the border of every shadow. Moreover, when temperature is measured, there exists the eminently practical problem of deciding how much weight to allocate to sample points which might be so unlucky as to fall in such hostile and inconvenient places as on an iceberg in the middle of the ocean or at the center of a volcano. The human decisions necessary in such circumstances inevitably lead to a degree of arbitrariness which might well explain the “observed” 0.6 degree C per century (or whatever it is) temperature increase. Incidently, two icebergs of the same surface area may have considerably different volumes and, hence, correspondingly different thermodynamic contributions but that is another issue entirely.

In summary, there exists the possibility that discrepancies in the measurement of global mean surface temperature may be measure-theoretic in origin, and, therefore not amenable to correction by, say, increases in sample size and quality. Such bias is intrinsic to the situation and independent of the measurements of even the most conscientious climatologist. A finite sample of points cannot, reliably estimate a distribution of unknown form over a set of positive measure. Similarly, surface measurements cannot reliably estimate an unknown volume distribution. I do not, for one second, doubt that climatologists have spent much time carefully discussing these issues and refining their sampling procedures. What I would like to know is whether or not any conclusion has been reached which does not rely on an ad hoc model assumption? If the answer is no, then we may well be justified in allowing the Chinese or other interested parties to adjust the level of significance by a factor of 10.

**Warmist**: You’re still not providing concrete examples of whether the problems you raise are actual problems, or just the kind of 0.001% error that we all ignore. At the risk of seeming repetitive in my last couple of responses, to answer the final question you pose in your comment please take a look at this RealClimate article posted in early July. In it you will get a sense for how these issues are dealt with.

**Philip Pennance**: Your request is again ambiguous and itself raises more questions. Problem for whom? Can a problem be a “non actual” problem? The answers to your request clearly vary according to the “problem”. For example, one concrete issue I raised was the “problem” of independent observers reconciling their respective data sets and the non uniqueness inherent in the treatment of the discrepancies. Is this an “actual” problem? Certainly. According to CCSP Synthesis and Assessment Report on Tropospheric temperature trends: https://www.climatescience.gov/Library/sap/sap1-1/finalreport/sap1-1-final-chap6.pdf

“There remain differences between independently estimated temperature trends for the surface, troposphere and lower stratosphere, and differences between the observed changes and model simulations, that are, as yet, not fully understood…”

“…The main lesson learned from this Report is that great difficulties in identifying and removing non-climatic influences from upper-air observations have led to a very large spread in trend estimates… “

Is this an “ignorable” problem?

That will depend on much argumentation concerning the possible causes and magnitude of the discrepancy —a technical question for climatologists. One thing that cannot be applied in this context is the concept of percentage error. Temperature is an ordinal variable so talk of percentage errors, as in your comment, can be meaningless in this type of analysis.

I leave you with a “non actual” problem which has some features in common with an actual problem.

Imagine a spherical planet, which we shall call X, that has only two weather stations. Suppose that for many centuries the two stations both record a stable temperature of 16° X. Suppose that in a certain year one station records an average temperature of 0° X and the other 36° X yielding a mean of 18° X Do you consider this to be evidence of global warming on the planet X? Do you consider this to be evidence of an increase in mean global surface temperature on the planet X? If so, at what level of significance? What assumptions are needed to calculate a level of significance?

**Warmist**: Philip, I don’t want to imagine a spherical planet. I want to imagine a real planet, and I want to understand how a real planet works. You are looking for “mathematical proofs”, but I fear you are looking for something that no one in the debate considers important to the matter at hand. Climate science is necessarily empirical and guided by our physical understanding of how the world works. Hundreds of studies have been done to see how good local measurements of temperature actually are, and whether they provide a sufficient proxy for the spatial and temporal patterns of temperature that we can safely say are found not on Planet X, but on this planet.

Climate science does not employ the logic of pure mathematics (contrary to your wish), but the logic of engineering. An engineer does not look for absolute precision, but for metrics to fall within certain tolerances, often with a safety margin. Similarly, climate scientists look for effects that are large and of first order, and while they fuss with the 0.001% issues as a matter of thoroughness, the broad picture is painted with the colors of 30%, 40% and 50% issues. Since temperature trends are broadly smooth over the surface of the earth at the spatial scale of relevant climate processes (i.e., tens and hundreds of miles), your issues are ultimately of the 0.001% variety.

The bottom line is that, yes, not all is fully understood. But to claim from the outside that because some aspect of a process (such some of the observed anomalies in upper and lower atmospheric temperature trends) is not yet understood signifies a deep pathology in the discipline as a whole is the novice’s mistake. Grad students get swept up in such fancies, only to later realize that they’re fussing over 0.001% issues.

**Philip Pennance**: For convenience my reply to each comment is in square brackets.

Philip, I don’t want to imagine a spherical planet. I want to imagine a real planet, and I want to understand how a real planet works. [**Einstein often used such thought experiments to shed light upon complicated situations.** ]

You are looking for “mathematical proofs” … [**Did I ask you for a proof? No, I merely posed an interesting question and asked what could be meaningfully said.**]

but I fear you are looking for something that no one in the debate considers important to the matter at hand. [**In my experience, universally negative statements of this form are frequently false.**]

Climate science is necessarily empirical and guided by our physical understanding of how the world works. [** All sciences must necessarily take account of the empirical. But most sciences like classical mechanics do not eschew mathematical proof when it is possible.**]

Hundreds of studies have been done to see how good local measurements of temperature actually are, and whether they provide a sufficient proxy for the spatial and temporal patterns of temperature that we can safely say are found not on Planet X, but on this planet. [**And as the reference in post 57 shows, there are many open issues.**]

Climate science does not employ the logic of pure mathematics (contrary to your wish), but the logic of engineering. An engineer does not look for absolute precision, but for metrics to fall within certain tolerances, often with a safety margin. [**Math, science and engineering are based on Aristotelian logic. I do hope that Climate Science has not abandoned the standard logic.**]

Similarly, climate scientists look for effects that are large and of first order, and while they fuss with the 0.001% issues as a matter of thoroughness, the broad picture is painted with the colors of 30%, 40% and 50% issues. [**It is physically meaningless to talk about percentages when talking about temperature. It is an ordinal variable.**]

Since temperature trends are broadly smooth over the surface of the earth at the spatial scale of relevant climate processes (i.e., tens and hundreds of miles), your issues are ultimately of the 0.001% variety. [**This seems tautological. They are smooth after considerable mathematical processing and filtering to make them smooth.**].

The bottom line is that, yes, not all is fully understood. But to claim from the outside that because some aspect of a process (such some of the observed anomalies in upper and lower atmospheric temperature trends) is not yet understood signifies a deep pathology in the discipline as a whole is the novice’s mistake. Grad students get swept up in such fancies, only to later realize that they’re fussing over 0.001% issues. **[But notice that I did not insult climatology by saying that there is a deep pathology in the discipline so your bottom line is both moot and unwarranted. ] **

**Warmist**: Philip [59], I find it difficult to respond to “fisks” (i.e., point-by-point responses in serial form) so I’ll address a couple of your points and we’ll go from there.

First, one can talk of percentages when speaking of the causal components of a fixed effect. For instance, one could say that 25% of the increase in temperature over the last 100 years is due to an increase in incoming solar radiation, while the remaining 75% is due to increases in trace greenhouse gases. It is this kind of percentage I’m talking about. A 50% effect is a big one, while a 0.001% effect is indeed minor and can be left out (much as an engineer would hope that her built-in safety margin never be overcome by departures in the large effects, let alone the small ones).

Second, regarding “proof”, I think where we’re getting ourselves into trouble (me mostly) is in the fact that science never actually deals in proofs, only in the synthesis of evidence and analysis which leads to theory. So, I thought you were asking for some kind of abstract “proof” that discontinuities in temperature do not exist (they do), or that a small number of measurements can accurately represent trends in much larger areas (they can). If you did in fact mean “proof”, then I’m afraid I can’t help you, but if you want evidence, then …

Third, regarding tautology, there is good evidence supporting the use of small numbers of measurements to represent much larger ones. Temperature stations, both urban and non-urban, are present across the United States, as elsewhere. Across the spatial scales at which climate-level processes occur (that is, the scale at which vortices and eddies in the atmosphere are realized; for example, the hundreds of miles across the width of tropical storms), significant discontinuities in those temperature records exist but they are not important. For example, the temperature in the air passing over a ridge line will drop as its pressure drops on the leeward side, but this is discontinuity that is MUCH, MUCH smaller than the large-scale atmospheric mixing that drives long-term changes in climate. Hence, local discontinuities like that over a ridge line can be safely ignored. Does one monitor the contour of the Pacific coastline foot-by-foot to determine how to navigate from Los Angeles to San Francisco by sea, or how the Humboldt current moves from Antarctica to southern Peru? No. Conversely, is precise measurement of the coastline in local areas important for understanding how organisms live in the tidal zone? Yes. The level of spatial precision needs to match the scale of the process in question. Processes at other scales are provisionally ignored since one can calculate that they would be unimportant at the scale in question. This is Aristotelean logic as applied to the process of theoretical simplification, and is a perfectly valid and commonly applied tool in all sciences.

Getting back to temperature, it is well known that temperature gradients over the spatial scales relevant to climate processes are, in fact, quite smooth, in part because those very same climate processes integrate the effect of local variation in temperature. In other words, it’s not the computer that does the smoothing, but the climate itself.

Please let me know if I have failed to respond adequately to your points. I confess that much of what you have written I have found confusing, but it could just be your use of the word “proof.” Much of the applied sciences register little interest in such things, which is why I was hesitant to engage you on something that is not especially important (if at all) to the discipline. Again, apologies for any misunderstanding.

**Philip Pennance**: You claim that “First, one can talk of percentages when speaking of the causal components of a fixed effect. For instance, one could say that 25% of the increase in temperature over the last 100 years is due to an increase in incoming solar radiation.”

On the other hand, in a recent paper, Does a Global Temperature Exist? J. Non-Equil. Thermod. 32, 1-27 (2007), (https://www.uoguelph.ca/~rckitri/research/globaltemp/GlobTemp.JNET.pdf), Essex, McKitrick, and Andresen: claim to show that the mean temperature is not physical, and that there may be many valid ways of computing a mean which will give different trends. If they are correct in this assertion, then trying to apportion percentage causes to, say, an increase in average global temperature, does not seem to make physical sense.

The above remark also applies to your later comment

“…it is well known that temperature gradients over the spatial scales relevant to climate processes are, in fact, quite smooth, in part because those very same climate processes integrate the effect of local variation in temperature. In other words, it’s not the computer that does the smoothing, but the climate itself.”

**Warmist**: Not being completely up to speed on how temporal trends are calculated from spatial data, I can only say that I’m sure if I were working on it, I would be worrying about it.

But I think we’re confusing something important here—no one I know is saying that mean temperature is a “real” thing, something that everyone everywhere experiences the same at all times. I pointed this outmean temperature is a diagnostic, not a prognostic.

Perhaps what we’re waiting for is a time when no matter how you calculate the mean from spatial data, current mean temperature is inarguably higher than it has been since the pre-industrial era, and in a way consonant with the physics of the climate.

BTW, I found the link you posted to go to a nowhere place on the web, but that this link (pdf) gets you the paper. Also, as usual, actual climate scientists with actual climate science jobs have honest rebuttals with actual arguments to the paper here and here. As pointed out by Rasmus Benestad (and let me allow his words to reflect my own impressions of your points), “The paper doesn’t bring any new revelations – I thought that these aspects were already well-known.”

**Philip Pennance**: You wrote: “But I think we’re confusing something important here—no one I know is saying that mean temperature is a “real” thing, something that everyone everywhere experiences the same at all times. I pointed this out mean temperature is a diagnostic, not a prognostic.” I never said (nor do I believe) “that mean temperature is a “real” thing, something that everyone everywhere experiences the same at all times.” Nor did Essex, McKitrick, and Andresen, nor indeed did anyone else that I know make such a claim. Indeed, I do not understand what you mean by “real” in this context. It does follow, from the axioms of mechanics (for a Galilean space time), that a mean height transforms to a mean height under the allowed changes of length scale. However, nothing in the axioms of classical thermodynamics permits us to prove the corresponding statement for temperature. This is the essence of the result by Essex et al. Whether or not “mean temperature is a diagnostic” is too vague and context laden a statement to assign a truth value. For example it is often false that mean temperature is a diagnostic of heating or cooling. Take ice at 0°C and heat it to form water at 0°C. There no change in the average temperature so the heating can not be diagnosed.

You say that “actual climate scientists with actual climate science jobs have honest rebuttals with actual arguments to the paper.

Actually, the blog sources you quote have more in common with a lynch mob than an appraisal by referees in a scientific journal. On the one hand you assert that the paper of Essex et al. has been rebutted and at the same time you quote Benestad who claims, providing no references, that “these aspects are well known.”. Thus the critics appear to contradict each other. There may well be non essential errors in the paper by Essex et al., but none that I can find which refutes their main thesis.

**Warmist**: There is no contradiction. Sometimes critics of an idea will raise criticisms that aren’t relevant. For instance, you point out that heating through a phase change doesn’t change the temperature. The proper response might be “duh” (with all respect). You present a well known fact that is already well incorporated in our understanding of climate physics. Likewise, if you’re a climate scientist and you read somewhere in a paper that how you calculate a mean will affect the mean value, you say “duh”. When you then hear from the same people that this is somehow a criticism of long-standing results, even though no actual analysis of the data itself was performed by the authors, you get a bit indignant. This was Benestad’s response. It is up to Essex et al. to show that calculating the mean values differently will actually affect the results, and they have to do this by actually calculating the means using real data. This is called constructive science, but like many critics (whether their criticism is of movies, books, evolutionary biology, or global warming) they fail to do little actual work themselves, much like the Scholastics did before Francis Bacon threw the book at Aristotle with his Novum Organum. The Essex paper was rebutted by showing that their points were either irrelevant to the discussion at hand or “well-known” and thus not really controversial. It is now on the shoulders of Essex et al. to go back through the data and show that a different mode of calculation will yield a different result. My guess, based on the kind of work they publish and on their preferred mode of criticism, is that they probably won’t bother to do that.

You may think the folks over at RealClimate to be a “lynch mob”, but if you would explore further you will see that they entertain all matter of criticism of their ideas, including criticism from folks like McKitrick and McIntyre (the authors of the “hockey-stick” critique and one of them a co-author of the Essex paper). For instance, please check out the comment section to the latest RealClimate post on the 1934 broohaha. You’ll see that Steve McIntyre (the author of the letter to the NASA GISS scientists regarding the error in the US data) gets involved in the comments at around comment 206. He is treated with respect, argument for argument.

Also, to directly respond to your criticism of temperature scales, I’m afraid you’re quite wrong. Small variations in temperature on an absolute scale (i.e., Kelvin) can be treated as linearly scalable, much as length scales can. This is done all the time in applied mathematics dealing with models of heat flux, and it has been done so for at least a hundred years. Also, as a matter of convenience, partitioning those small changes in terms of percentages is perfectly acceptable and is done all the time.

What do I mean by real? I will pose my definition in Essex et al.’s own words (p. 2):

While that statistic is nothing more than an average over temperatures, it is regarded as the temperature, as if an average over temperatures is actually a temperature itself, and as if the out-of-equilibrium climate system has only one temperature. But an average of temperature data sampled from a non-equilibrium field is not a temperature.

I completely agree with them, as would anyone working in the field. Global temperature is not a “real” thing because it isn’t experienced anywhere by anyone. It is a diagnostic of the state of globe’s temperature at any one time, and is never meant to represent temperature at all points, just as the mean height of Americans is not meant to represent the actual height of all Americans. (And, yet, we’re now the 2nd tallest country in the world, recently losing our 1st place status to the Netherlands. Does that mean that I’m shorter than all Dutch? Of course not.)

The bottom line is that the criticism presented by Essex et al. is trivial. Of course global temperature as a single number does not really exist. It’s an index. Anyone who might have thought otherwise would have to be considered mightily daft.

But here’s where I get pretty annoyed with Essex et al. (again, p. 2):

The resolution of this paradox is not through adoption of a convention. It is resolved by recognizing that it is an abuse of terminology to use the terms “warming” and “cooling” to denote upward or downward trends in averages of temperature data in such circumstances. Statistics might go up or down, but the system itself cannot be said to be warming or cooling based on what they do, outside of special circumstances.

… or (p. 3):

Debates as to which is the correct one are fundamentally false, with no correct resolution.

Welcome to the road to postmodernism. Hey man, whatever makes you feel good, but are they really trying to suggest that if the globe did become significantly and unequivocally warmer than it is now we wouldn’t be able to measure it because there would be a dispute about how to measure it? Gimme a break; these guys are the worst sort of armchair scientists. As someone who works hard in a lab all day, let me say that I feel completely justified in demanding that they get off their keesters and do some actual work.

**Philip Pennance**: you say “Sometimes critics of an idea will raise criticisms that aren’t relevant. For instance, you point out that heating through a phase change doesn’t change the temperature.”

This does seem relevant in the context of your remark [70] that “mean temperature is a diagnostic, not a prognostic.” In fact, if you had not made a scientifically ambiguous claim, I would not have wasted your time and mine citing a simple counterexample. I was never under the delusion, as you assume, that climatologists were unaware of utterly trivial facts about the melting of ice. I did not even mention the concept of phase change. This was inserted by you to add credibility to your insinuation. I merely confined myself to providing a simple counterexample. Many other examples could have been given, but this would have been unnecessary. In any case, your error is not relevant to the discussion so let’s move on.

You also state that: “Essex paper has been rebutted by showing that their points were either irrelevant to the discussion at hand or “well-known” and thus not really controversial”. In fact, the crucial point for climatology seems to be the claim that, from a set of temperature readings alone, one can deduce neither an increase nor a decrease in mean global temperature. Has this point really been rebutted? If it has, then your next remark:

“it is up to Essex et al. to show that calculating the mean values differently will actually affect the results.” is redundant. After all, if Essex et al. do succeed in this task, would this not rebut the rebuttal?”

The following statements can be found in the references you cited:

“When I first read this paper I thought it was a joke”

“I believe it is only by going to totally stupid definitions of “average” that they are going to see any significant differences in global temperature trends”

“Their fundamental error is that they wrote the paper.”

“You can find details of procedure in Ref 1 and 2 of the Essex, et al. paper. Let us examine why this is done”. Hey, who really cares why this is done?”

Do you really believe that scientific etiquette requires the authors of a published paper, or anyone, to pay attention to hostile criticism from blog posts such as the above? A natural way of handling an alleged scientific error is a polite letter to the editorial board of the journal outlining the problem and requesting either a correction, a refutation. or a retraction as the case warrants. Perhaps customs are different in Climatology.

Rather than waiting for Essex et al. to show that calculating the mean values differently will actually affect the results, let us return to my previous analogy. In the light of your previously stated aversion to thought experiments I will avoid discussing hypothetical spherical planets and alter the situation slightly.

Historians of science inform us that in the early days of “climatology”, a wide variety of different thermometric fluids were in use. Imagine that two early climatologists simultaneously measure the temperature at their respective locations and obtain an arithmetic mean of 16°X where X is the temperature scale defined by their common thermometric liquid. Suppose that at a given time later, both take new simultaneous measurements; one obtaining 0°X and the other 36°X for a mean of 18°X. In this case the arithmetic mean has increased by 2°X. Imagine that many years later, a third climatologist then transforms these measurements to °C. Suppose (temporarily) that the change of scale between °X and °C is given by the square root function which is continuous and strictly increasing. The initial average temperature reading of 16°X converts to 4°X. The final readings of 0°X and 36°X convert respectively to 0°C and 6°C yielding an arithmetic mean of 3°C —a drop of 1°C. This shows that the same data set can be compatible with an increase in the arithmetic mean on one temperature scale and a decrease on another. Moreover, there is nothing special here about the square root function. One can find infinitely many strictly increasing continuous functions, and compatible with the axioms of classical thermodynamics, which would illustrate the same behavior. Thus, to obtain the results of Essex at el., at least in classical thermodynamics, it is not even necessary to consider what the critics cited above call stupid definitions of “average”. Even the arithmetic mean is subject to this pathology when non affine changes of scale are considered. Thus, Essex et al.’s paper appears to be correct on this important point and the above example provides another refutation of the universality of your statement [70] that “mean temperature is a diagnostic”.

**Warmist**: Philip, regarding “scientific etiquette”, let me just say that blogs are blogs, and journals are journals. The bottom line is that Essex et al.’s paper is getting a hearing in public. They’re big boys, just like you. The other bottom line (yes, there are two bottom lines) is that I have no idea who might have reviewed their manuscript when it was delivered to the Journal of Nonlinear Thermodynamics, but I am quite sure that it was received with decorum and courtesy. Either way, taken together, both of the responses to the paper that I linked to raise all the important issues, including the one about the paper being an utterly trivial exercise, since everyone already knows what they’re writing about.

You were offended by my “insinuation” that you somehow weren’t aware of phase changes. I didn’t mean it that way, but I apologize anyway. The point of my quip was that Essex et al. never made the argument that you do about “warming” and “cooling” when you wrote:

For example it is often false that mean temperature is a diagnostic of heating or cooling. Take ice at 0°C and heat it to form water at 0°C. There no change in the average temperature so the heating can not be diagnosed.

They are basing their argument on the problems associated with generating an agreed upon method of calculating a mean global temperature, which I think you understand (see below). You were making a different point from what they were, and so I felt the need to swat it as not relevant to conversation. But enough with relevance! With them, I agree that no such temperature exists; it’s an index. But their error is to solipsistically “give up” on the idea of ever generating an index for change in global temperature. It’s almost as if they would balk if I told them the surface of Venus was hotter than the surface of the Earth: “But wait,” they would say, “you have no agreed upon method of measuring the mean temperatures of these two planets, therefore leave me in peace while I walk my dog and think about new ways to confound the public.”

Obviously, at some point, a difference in mean temperature between now and some point in the past will become so great that it won’t matter how you calculate the mean. And this is the kernel of Essex et al.’s difficulties: would it matter to what we know now if they calculated the Earth’s mean temperature in any one of the hundred or so possible ways that they allude to in their paper? The motto of constructive science is “go and see.” Instead, they sit and snipe. In fact, we really actually must wait “for Essex et al. to show that calculating the mean values differently will actually affect the results,” or at least for someone else to do so. Until then, they will have failed to meet the high standards of the “engineering” logic of climatology which requires that one do the work necessary to test one’s hypothesis. They present an hypothesis, that how one calculates the mean will matter to the result, but then they don’t test it. What kind of science is that?

Finally, you write:

Suppose (temporarily) that the change of scale between °X and °C is given by the square root function which is continuous and strictly increasing.

Whaaaa? Where could this possibly be leading. (Are you thinking of the square-dependence of mean free kinetic energy on temperature?) Wait, I will be patient … you go on:

Even the arithmetic mean is subject to this pathology when non affine changes of scale are considered.

Sorry, but since when has any temperature scale in the last 100 years been transformed by a square root function (and then not transformed back)? I call straw man.

**Philip Pennance**: You say [78] that: “With them, I agree that no such temperature exists; it’s an index. But their error is to solipsistically “give up” on the idea of ever generating an index for change in global temperature.”

If one fixes a temperature scale (say Kelvin) for all time, then one certainly has an index of mean global surface “Kelvin temperature” average. As long as everyone sticks to that scale, or to scales in an affine relationship with that scale (as is done in practice), then no problems arise. On that point I agree with you. However, the fact that mean temperature does not behave the same way as mean length should not be dismissed as a “theoretical” irrelevance” It does impose the constraint that during non linear smoothing and filtering operations on temperature data, all transformations used must preserve the mean with respect to the chosen fixed scale. I will presume, for the sake of argument, that this issue is handled correctly. However, as I appointed out, even if a scale is fixed an increase in average temperature can be consistent with cooling (net heat loss) in a non homogeneous body such as the earth. Thus our agreed upon fixed Kelvin index is not diagnostic of heating or cooling without supplementation from other data as illustrated by the simple example above.

Let me address your point that small variations in temperature on an absolute scale (i.e., Kelvin) can be treated as linearly scalable, much as length scales can. In the case of length, this fact follows from a continuity assumption and the fact that the length of a concatenation of two bodies is the sum of the lengths. In classical thermodynamics, there is no analogous concatenation operation for temperature and so there is no basis for this assumption. It is not an argument that “this has been done so for at least a hundred years”.

**Philip Pennance**: Note that by the term “linearly scaleable” in my previous post I mean “necessarily linear scaleable” in the sense that scales of length must transform linearly, whereas, in the case of temperature, the class of allowable scale changes includes strictly increasing continuous functions which are non linear. There is no disagreement if you are merely saying that a linear or affine transformation is a valid change of temperature scale.

**Warmist**: If what you’re saying represents the entirety of your response, then I can’t imagine what we are still disagreeing about. To go further, I wonder if you are willing now to consider that the Essex et al. (2007) paper provides no non-trivial contribution to the question of whether we are experiencing a global warming trend, or that some index must be used (contrary to their point) lest we sink into a morass of solipsistic and anti-scientific thinking, or that science from the “armchair” on the subject of what is actually happening in the world is of little value unless it is put into practice. Essex et al. have much work yet to do. Who knows? Perhaps they’ll show that among a set of reasonable indices of global mean temperature, how the index is calculated makes a big difference. I look forward to their results.

**Philip Pennance**: From a mathematical perspective some conclusions in the paper of Essex et al. do seem to me unsurprising. The fact that the direction of a temperature trend can change if a geometric mean rather than an arithmetic mean is used is a simple consequence of the arithmetic-geometric mean inequality. Actually, a change to the geometric mean is equivalent to retaining the arithmetic mean and making a logarithmic change of temperature scale because, under such a scale change, the arithmetic mean maps to the geometric mean. That such a transformation is valid would follow from the ordinality of temperature and the fact that the class of admissible changes of temperature scale includes strictly increasing, differentiable functions such as the logarithm.

However, I am somewhat worried by the same philosophical issue that appears to bother Essex at el. Imagine now that statistical mechanics and quantum mechanics had never been discovered and that all thermometric scales instead were based on a variety of “designer” fluids. There might be no reason under these circumstances to prefer one thermometer over another and different scales including the Kelvin scale would have equal status. Moreover, there is no physical result, expressible in the Kelvin scale, that could not be re-expressed (via a bijective scale change) in terms one of these other scales. In this sense at least, Kelvin is not ”all powerful” but, rather, when compared to other scales ”equally powerful”. In this non existent but possible world, different scales would indeed yield different temperature trends and even trends of opposing signs. In the real world, we “privilege” — please excuse this horrible “postmodernism”— the Kelvin scale but that decision in itself is not unique. Even in the context of statistical mechanics no error could result if another scale were chosen and all formulae rewritten to compensate. So is all of this really irrelevant in the real world of actual climate change? Let us look at what the critics say.

Many of the critics who claim that the paper of Essex et al. contains well known and trivial results, themselves make questionable claims. For example, in an article at realclimate.org it is claimed that the work of Essex et al. is

“irrelevant in the context of a climate because CO2 affects all surface temperatures on Earth.”

Obviously, if temperatures rose everywhere there would be a temperature rise on all valid scales. However, anthropongenic CO2 does not effect temperatures at all points equally. Since the system is inhomogeneous, some temperatures can be expected to rise and some fall. Even if we ignore the slow diffusion and mixing of CO2 and assume that temperatures rise everywhere simultaneously, but at distinct rates, then different (non affinely related) temperature scales will still yield different rates of increase for the mean.

The same article also states that: “Temperature itself can be inferred directly from several physical laws, such as the ideal gas law, first law of thermodynamics and the Stefan-Boltzmann law, so it’s not the temperature itself which is ‘unphysical’ ”

All this statement seems to say is that there exist a variety of thermometers —which is obvious. This argument does not disprove the claim that average surface temperature has no physical meaning. In fact, Essex et al. did not, as far as I know, claim that temperature at a point was unphysical or could not be inferred by a variety of devices or proxies.

Other critics claim (without any proof) that temperature is an interval, as opposed to ordinal variable. Although this same statement can be found in many statistics texts, none I can find offers anything like a proof. Thus the possibility exists that this objection is based upon a non existent theorem —at least, a theorem which was not in my thermodynamics text 40 years ago.

Yet another critic in the same realclimate.org page states that:

“Ultimately, it is energy that counts. Any global measure that is consistent with the trends in either atmospheric, ocean or atmosphere-ocean energy is potentially useful in describing what is happening to these parts of the Earth system. Thermodynamic temperature determined as simple weighted averages dealing with heat capacities and masses works as well as anything else for this.”

I agree that energy is what really counts but the last sentence is false. A single net heating can manifest itself in many different manners. For example, as an increase in volume isothermally or as an increase in temperature at constant volume or as a combination of volume and temperature change. Temperature averages, therefore, cannot on there own determine energy uniquely, even using a “weighted average dealing with heat capacities and masses”. More generally, 0.6°K per century increase in surface temperature does not on its own imply a net heating, even when a fixed temperature scale such as Kelvin is used.

It is ironic that critics of the paper of Essex et al., who claim it is both trivial and unoriginal, themselves make questionable and ambiguous claims.

**Footnote: No further replies were received. A year or so later, the warmist deleted his contribution.**